Set Theory and the Foundation of Mathematics

Axiomatic set theory and the foundation of mathematics

April – August 2019, Kobe

渕野昌 (Saka´eFuchino) [email protected]

main updates: 19.07.17(We09:24(JST)) 19.06.29(Sa00:35(JST))

Last updated on: 2021 年 04 月 08 日 (09:20 (JST))(0)

Contents 1. Na¨ıve axiomatic set theory ...... 1 1.1. Zermelo Set Theory and the Axiom of Choice ...... 2 1.2. Zermelo-Fraenkel Set Theory ...... 4 2. Predicate logic ...... 4 2.1. Predicate logic in meta-mathematics ...... 4 2.2. Axiomatic set theory over predicate logic ...... 9 2.3. The deduction system K∗ ...... 12 2.4. Predicate logic in Z and the Completeness Theorem ...... 18 2.5. Some semantic proofs of proof theoretic assertions ...... 18 3. Incompleteness Theorems and Speed-up Theorems ...... 19 3.1. Ehrenfeucht-Mycielski Speed-up Theorem ...... 19 3.2. Diagonal lemma ...... 23 3.3. G¨odel’sSpeed-up Theorem ...... 29 References ...... 31

1 Na¨ıve axiomatic set theory

naive In the following, The axiom system ZFC (Zermelo-Fraenkel Set Theory with Axiom of Choice) of set theory is formulated with certain redundancy. This formulation is chosen

(0) The present text is a lecture note of the course I gave in the summer semester 2019 at Kobe university. The text is further extended and worked-out when I gave a similar course in Katowice in October 2019. The most up to date version of this text is downloadable as: http://fuchino.ddo.jp/kobe/logic-ss2019.pdf At present this text is still a work in progress. It will be constantly updated during and after the courses in 2019. Any comments and/or questions in connection with this text are appreciated, and are to be sent to the email address above.

1 on purpose so that some important subtheories of ZFC are subsets of the system. The axioms of ZFC are built on the single predicate “∈” where “x ∈ y” means x belongs to y as an element. We do not introduce a predicate for expressing “x is a set” since everything is a set in set theory.(1) If we say “for all x ...” in an axiom of set theory, what is meant is “for all set x, ...”.

1.1 Zermelo Set Theory and the Axiom of Choice zermelo The following axioms make the axiom system of Zermelo Set Theory (Z) which is a subset of the axiom system of ZFC introduced in the next subsection.

(Extensionality) If we have u ∈ x if and only if u ∈ y for any u, then x = y.

(Empty Set) There is x such that u ̸∈ x for any u.

By the Axiom of Extensionality, the set x as in the Axiom of Empty Set exists uniquely. We denote this set by ∅.

(Pairing Axiom) For x and y there is z such that, for any u, u ∈ z if and only if u = x or u = y.

Again by the Axiom of Extensionality, the set z as in the Pairing Axiom for each x and y is unique. We shall denote this unieque z up to x and y by {x, y}. If x = y, we write {x, x} = {x} and call it the singleton x.

(Axiom of Union) For any x, there is y such that, for any u, u ∈ y if and only if u ∈ z S for some z ∈ x (notation: y = x).

The next axiom is actually an axiom scheme in the sense that, for each property Φ, an instance of the axiom is made.

(Axiom of Separation) If Φ(·) is a property expressed by using only “∈” and “=” then for any x there is y such that u ∈ y if and only if u ∈ x and Φ(u) holds.

The set y in the Axiom of Separation is often denoted by y = {u ∈ x : Φ(u)}. S By Pairing axiom and Axiom of Union, we can construct {x, y} for any x and y. This set is denoted by x ∪ y. Clearly we have u ∈ x ∪ y if and only if u ∈ x or u ∈ y for any u.

(Axiom of Inﬁnity) There is x such that

2 (2) (1.1) ∅ ∈ x and y ∪ {y} ∈ x for any any y ∈ x. naive-a

(Power-Set Axiom) For any x, there is y such that, for any u, u ∈ y if and only if all elements of u are elements of x.

The set y in the Power-set Axiom is denoted by y = P(x) and y is said to be a it power- set of x. The relation “u ∈ y then u ∈ x for all u” is denoted by y ⊆ x and y is said to be a subset of x. With this notation, we can write P(x) = {y : y ⊆ x}. The construction of power-set seen in this way is outside the scope of the Axiom of Separation since a set which would delimit the range of y in {y : y ⊆ x} (for example P(x)) can only be obtained using the Power-Set Axiom. Ordered pair of sets a and b can be treated as the following set.

(1.2) ⟨a, b⟩ = {{a}, {a, b}}. naive-0

. This deﬁnition is a reasonable one beacuse of: L-naive-0 Lemma 1.1 For any a, b and a′, b′ ⟨a, b⟩ = ⟨a′, b′⟩ if and only if a = a′ and b = b′.

L-naive-1 Lemma 1.2 For any a and b the is c such that, for any u, u ∈ c if and only if there are v ∈ a and w ∈ b such that u = ⟨v, w⟩.

Proof. Recalling (1.2), such c can be introduced as

(1.3) {u ∈ P(P(a ∪ b)) : u = ⟨v, w⟩ for some v ∈ a and w ∈ b}. naive-1

There is such a set because of the Axiom of Power-set and the Axiom of Separation.

(Lemma 1.2)

For a, b, the set c as in Lemma 1.2 is uniquely determined. We call such c a (Cartesian) product of a and b, and denote it by a × b. For a, and b, f ⊆ a × b is said to be a function from a to b, if, for any v ∈ a, there is a unique w ∈ b such that ⟨v, w⟩ ∈ f (notation: f : a → b). For v ∈ a the unique w ∈ b with ⟨v, w⟩ ∈ f is denoted by f(v) = w and called the value of f at v. S (Axiom of Choice) For any x such that ∅ ̸∈ x, there is a function f : x → x such that f(u) ∈ u for all u ∈ x.

(1) That is, except proper classes which are just reformulations of properties of sets and hence not objects in set theory.

(2) Not that the set y ∪ {y} is the set of all elements of y and y itself. In [Zermelo 1908], the Axiom of Infinity was formulated with (1.1) replaced by “∅ ∈ x and {y} ∈ x for any y ∈ x”. The original form of the Axiom of Infinity is equivalent to our Axiom of Infinity in ZF but the equivalence is not provable in the Zermelo Set Theory Z. The formulation of the Axiom of Infinite with (1.1) is choosen here to make the collecgtion of all natural numbers ω a set under the standard definition of natural numers due to von Neumann possible in Z.

3 Axiom of Choice is abbreviated as AC. The axiom system obtained by adding the Axiom of Choice to the Zermelo Set Theory Z is denoted by ZC and called the Zermelo Set Theory with AC.

1.2 Zermelo-Fraenlel Set Theory zermelo-fraenkel The axiom system of Zermelo-Fraenlel Set Theory (ZF) is obtained by adding the following two axioms to the axiom systemo of Zermelo Set Theory. The ﬁrst one of the two additional axioms is actually an axiom scheme like Axiom of Separation:

(Replacement) For any x, if Φ(·, ·) is a property expressed by using only “∈” and “=” such that, for any u ∈ x there is the unique v such that Φ(u, v) holds, then there is y such that v ∈ y if and only if Φ(u, v) for some u ∈ x (notation: y = {v : Φ(u, v) for some u ∈ x}).

The signiﬁcance of the next axiom of ZF becomes clear in connection with the theory of the transﬁnite induction.

(Foundation) For any non empty x, there is y ∈ x such that there is no y′ ∈ x with y′ ∈ y.

2 Predicate Logic

pred-logic 2.1 Predicate logic in meta-mathematics meta The na¨ıve approach to the axiomatization of set theory we saw in the previous section was inacurate in many respects. One of the main problem was that we could not specify what are “set theoretic properties” in the definition of the Axiom of Separation and Axiom of Replacement. We want to re-introduce the axioms of set theory on the basis of the formal logic which will be introduced below. When we introduce the logic, we do it before introducing the axionms of set theory. We are in a world outside set theory. This means in particular that, at this stage, we cannot use the fictive notion of infinity, like the set of all natural numbers. If we write nevertheless some thing like “For n ∈ N, ...”, then this is merely abbreviation of the statement “For a concretely given expression (numeral) n of a natural number, ...”. The mathematics available at this stage must be the arguments on finite concrete objects, like symbols and finite sequences of symbols, with concrete operations, like basic operations on digits, operations on finite sequences (concatenation of sequences, for example), etc. This standpoint is often called meta-mathematics from which most

4 the following arguments in this section are done. In summary, the metha-mathematics is the world of finite sequences of symbols on an imaginary paper.(3) The notion of meta-mathematics was introduced by David Hilbert and he used the word ,,finitär“to describe the concrete operations on concrete objects. The word is then translated to the invented word “finitary” in English. This word “finitary” is very often confused with the word “finite”. All the objects treated in the finitary satnd point are at most potentially infinite and we do not assume the existence of infinite sets like the set of all natural numbers. In meta-mathematics, we try to establish general facts about concretely given objects, like a concretely given expressions for numbers, say 1, 2457, etc., or a concretely given sequence of symbols “abc”, “x ≡ y” etc., and concrete operations on these object. An infinite collection can be also concretely given. For example, an infinite “subset” S of N is concretely given if an algorithm is given according to which we can decide whether an arbitrary concretely given number n belongs to S. Thus, if we begin an argument declaring that something is “concretely given”, this signalizes that the following discussion is done in meta-mathematics and we are talking about concrete operations on the object which is executable (at least theoretically(4) ) once it is really concretely given. Actually, we are going to write this expression “something be concretely given” many times in the following, since we transfer back and forth between meta-mathematics and the mathematics inside a formal system, and since it would be awkward to say a phrase like “now we work again in meta-mathematics” each time when we return to a discussion in meta-mathematics. To define our formal system of predicate logic, which we call K∗,(5) , we first set the collection of symbols which are to be used in the system. The following symbols are mandatory for all instances of the formal theories:

(6) (2.1) variable symbols: ‘x0’, ‘x1’, ‘x2’,... meta-a-0 (2.2) logical symbols: ‘→’, ‘¬’, ‘∃’

(3) We say here an “imaginary paper” since we do not place any restriction on the size of the paper or on the physical feasibility of the operations done on the paper: The paper can be so large that we can write more symbols than the number of the elementary particles in the whole universe and operations undertaken on the symolbs written on the paper may be repeated more than the times the fastest CPU clock will tick from the begin of the universe till now.

(4) That is, if we ignore the physical limitations on time and resource for the execution of the manipu- lation.

(5) The choice of the name K∗ for the system we are introducing in the following is quite arbitrary except that the letter K is chosen because of the German expression ,,logischer Kalk¨ul“which means a logical deduction system.

(6) We assume that all variables are formally of the form “xi”. Although, for readability, we shall also use some other letters like ‘y’, ‘z’, ‘y′’‘z′’ etc. to represent variables.

5 (2.3) equality symbol: ‘≡’

(2.4) auxiliary symbols: ‘(’, ‘)’, ‘,’ meta-a-1

Some of the following symbols are used depending on the instance of the formal theory:

(2.5) constant symbols: ‘c0’, ‘c1’, ‘c2’ ,...

(2.6) function symbols: ‘f0’, ‘f1’, ‘f2’ ,... (7) (2.7) relation symbols: ‘r0’, ‘r1’, ‘r2’ ,...

We assume that all the symbols we introduced are distinct from each other. To be more concrete, and to make the translation of notions of meta-mathematics into set theory smoother, we assume here that all these symbols are pairs of numerals. To be concrete, let us assume that

(2.8) the variable symbols ‘x0’, ‘x1’, ‘x2’,... are the pairs ⟨0, 0⟩, ⟨0, 1⟩, ⟨0, 2⟩,... resp., (2.9) the logical symbols ‘→’, ‘¬’, ‘∃’ are the pairs ⟨1, 0⟩, ⟨1, 1⟩, ⟨1, 2⟩ resp., (2.10) the equality symbol ‘≡’ is the pair ⟨2, 0⟩, (2.11) the auxiliary symbols ‘(’, ‘)’, ‘,’ are pairs ⟨3, 0⟩, ⟨3, 1⟩, ⟨3, 2⟩ resp.,

(2.12) the constant sysmbols ‘c0’, ‘c1’, ‘c2’ ,... are the pairs ⟨4, 0⟩, ⟨4, 1⟩, ⟨4, 2⟩,... resp.,

(2.13) the function sysmbols ‘f0’, ‘f1’, ‘f2’ ,... are the pairs ⟨5, 0⟩, ⟨5, 1⟩, ⟨5, 2⟩,... resp., and

(2.14) the relation sysmbols ‘r0’, ‘r1’, ‘r2’ ,... are the pairs ⟨6, 0⟩, ⟨6, 1⟩, ⟨6, 2⟩,... resp.

We then ﬁx a computable function ar : N → N\{0} such that, for each k ∈ N\{0}, there are inﬁnitely many n ∈ N such that ar(n) = k and we assume the function sysmbol fn

(i.e. the pair ⟨5, n⟩) and the relation symbol rn (i.e. the pair ⟨6, n⟩) have the arity ar(n). A collection L of constant, function, and/or relation symbols is called a language. We fix a language for each theory we would like to develop over the logical system K∗. To specify a language, we often fix some finite index set I, J, K ⊆ N and write

(2.15) L = {ci, fj, rk}i∈I,j∈J,k∈K . meta-0

(7) We assume that a fixed arity (the number of the parameters the symbol accept) is attached to each of the function and relation symbols. We assume that we have unbounded number of function and relation symbols in stock for each given arity. Similarly to variables, these symbols are officially fixed as written here. In spite of this strict setting we often call some of these symbols differently to make the formal expressions more connected to the conventional notation. For example the binary relation symbol in the language of set theory which should represent the element relation is written as ε instead of rk for some k ∈ N.

6 Some/all of the index sets I, J, K may be empty. For example, in case of Zermelo-Fraenkel set theory in the non-na¨ıve axiomatization, which will be introduced in the next section, we use a language with a single binary relation symbol which should represent the element relation. This relation symbol (rn for some n with ar(n) = 2) is denoted as ε and the language of the (Zermelo-Fraenkel) set theory is denoted by Lε. Thus Lε = {ε}.

For a given language L, let SeqL be the “set” of all ﬁnite sequences of symbols in (8) (2.1) ∼ (2.4) as well as symbols in L. For a sequence s ∈ SeqL, ℓn(s) denotes the length of the sequence.

We deﬁne L-terms ⊆ SeqL recursively as follows:

(2.16) If x is a variable symbol, then the sequence of length 1 consisting of x is an meta-1 L-term;(9)

(2.17) If c is a constant symbol in L then the sequence of length 1 consisting of c is an meta-2 L-term;(10)

(2.18) If f is an k-ary function symbol in L and t0,..., tk−1 are L-terms, then f(t0, ..., meta-3 (11) tk−1) is also an L-term;

(2.19) Nothing else. meta-4

In case of the language Lε of set theory, since it does not have any constant or function symbol, Lε-terms are just sequences of length 1 consisting of a variable symbol. For an L-term t, free(t) denotes the set of all variable symbols which appear in t. If free(t) ⊆ {v0, ..., vk−1}, for some variable symbols v0, ..., vk−1 (each of these symbols is of the form xn for some natural number n), we write t = t(v0, ..., vk−1). Note that, by this notation, there may be some variables in the list v0, ..., vk−1 which do not appear at all in t. Such variables are sometimes called “dummy variables”. The possibility of dummy variables is included here on purpose to enable some details of induction proofs easier to formulate.

For a language L, the L-formulas ⊆ SeqL are now deﬁned recursively as follows:

(12) (2.20) If t0, t1 are L-terms then t0 ≡ t1 is an L-formula; meta-5

(2.21) If t0,..., tk−1 are L-terms and r is a k-ary relation symbol in L, then r(t0, ..., meta-6

(8) We treat the symbol SeqL similarly to N above. (9) Sometimes we write simply x to denote this sequence “x”.

(10) Also we write simply c to denote this sequence.

(11) With “f(t0, ..., tk−1)”, we denote the sequence obtained by concatenating all of these symbols and sequences in this order. The same convention will be used hereafter without mention.

(12) Similarly to the footnote (11) , we mean with t0 ≡ t1 the concatenation of sequence t0, symbol ≡ and sequence t1. We simply drop further notices of similar kind (e.g. in (2.21), (2.22), etc).

7 (13) tk−1) is an L-formula;

(2.22) If φ0 and φ1 are L-formulas then (φ0 → φ1) is also an L-formula; meta-7

(2.23) If φ0 is an L-formula, then ¬φ0 is also an L-formula; meta-8

(2.24) If φ0 is an L-formula and x a variable symbol, then ∃x φ0 is also an L-formula; meta-9

(2.25) Nothing else. meta-10

L-formulas of the form (2.20) or (2.21) are called atomic formulas. A variable symbol x in an L-formula φ is said to be free if it is not in the scope of a quantification “∃x” in φ (see for exact definition below). L-formula φ without free variables is called an L-sentence. We shall define the notion of subformula and free variable more precisely below. The intended reading of these formulas is as follows:

(2.26) t0 ≡ t1: “t0 and t1 are identical”. meta-16

r(t0, ..., tk−1): “t0, ..., tk−1 stand in the relation r”.

(φ0 → φ1): “φ0 implies φ1”.

¬φ0: “φ0 does not hold”.

∃x φ0: “there exists an x such that φ0 holds”.

For readability, we often use the following abbreviations:

(2.27) abbreviation actual sequence; explanations meta-17

t0 ̸≡ t1 ¬t0 ≡ t1

t0 f t1 f(t0, t1); for some binary function symbols f,

like “t0 + t1”, “t0 · t1”, etc.

t0 r t1 r(t0, t1); for some binary relation symbols r,

like “t0 ε t1”, “t0 < t1”, etc.

t0 ̸ r t1 ¬r(t0, t1); for some binary relation symbols r,

like “t0 ̸ε t1”, “t0 ≮ t1”, etc.

(φ0 ∨ φ1) (¬φ0 → φ1)

(φ0 ∧ φ1) ¬(φ0 → ¬φ1)

(φ0 ↔ φ1) “((φ0 → φ1) ∧ (φ1 → φ0))”; “∧” in the expression must be further resolved.

∀x φ0 ¬∃x¬φ0

∃x! φ0 “(∃x φ0 ∧ ∀x0∀x1 ((φ0(x0/x) ∧ φ0(x1/x)) → x0 ≡ x1))”;

x0 and x1 are variables not appearing in φ0, “∀” and “∧” in the expression must be further resolved.

(13) For binary relation symbol r we often write t0 r t1 instead of r(t0, t1). In particular, in case of Lε, we write x ε y instead of ε(x, y) see (2.27) for similar shorthands.

8 According to the intended interpretation of L-formulas (2.26), these abbreviations can be read as follows:

(2.28) t0 ̸≡ t1: t0 and t1 are not identical. meta-18

t0 ̸ r t1: r(t0, t1) does not hold.

(φ0 ∨ φ1): at least one of φ0 and φ1 holds.

(φ0 ∧ φ1): both of φ0 and φ1 hold.

(φ0 ↔ φ1): φ0 is equivalent to φ1.

∀x φ0: φ0 holds for all x.

∃x! φ0: there exists exactly one x satisfying φ0.

For an L-formula φ with ℓn(φ) = ℓ, let si denotes the ith symbol in the sequence φ for i < ℓ. For i < j ≤ ℓ Let φ⟨i,j⟩ be the subsequence “sisi+1 ··· sj−1” of φ. We deﬁne the set Subφ ⊆ {⟨i, j⟩ : i < j ≤ ℓ} of the positions of subformulas recursively as follows:

(2.29) If φ is an atomic formula and ℓ = ℓn(φ), then Subφ = {⟨0, ℓ⟩}; meta-11

(2.30) If φ is (φ0 → φ1) for L-formulas φ0, φ1, ℓ = ℓn(φ), ℓ0 = ℓn(φ0) then Subφ = meta-12 {⟨ ⟩ ⟨ ⟩ ∈ }∪{⟨ ⟩ ⟨ ⟩ ∈ }∪{⟨ ⟩} i+1, j +1 : i, j Subφ0 i+ℓ0 +2, j +ℓ0 +2 : i, j Subφ1 0, ℓ ; ¬ {⟨ ⟩ ⟨ ⟩ ∈ }∪{⟨ ⟩} (2.31) If φ is φ0 and ℓ = ℓn(φ), then Subφ = i+1, j +1 : i, j Subφ0 0, ℓ ; meta-13 ∃ {⟨ ⟩ ⟨ ⟩ ∈ } ∪ {⟨ ⟩} (2.32) If φ is x φ0 and ℓ = ℓn(φ), then Subφ = i2, j + 2 : i, j Subφ0 0, ℓ . meta-14

Suppose that x is a variable symbol and it is a kth symbol of an L-formula φ. Then we say that x appears freely in φ as the kth symbol of φ if there is no ⟨i, j⟩ ∈ Subφ such that i ≤ k < j, and φ⟨i,j⟩ is of the form ∃x ψ. For an L-formula φ with ℓ = ℓn(φ), let

(2.33) free(φ) = {x : x is a variable symbol and x appears in φ freely as kth symbol meta-15 of φ for some k < ℓ}. L ⊆ { } For an -formula φ, if free(φ) v0, ..., vk−1 , for some variable symbols v0, ..., vk1 , we write φ = φ(v0, ..., vk−1). L-formula φ with free(φ) = ∅ is called an L-sentence.

2.2 Axiomatic set theory over the predicate logic axiomatic-set- L Let ε be the language consisting solely of a single binary relation symbol ε. We can theory reformulate the axioms introduced in vernacular language of mathematics in Section 1 as a collection of Lε-sentences.

The following Lε-sentences make up the axiom system of Zermelo set theory. We say the axiom system, since we have to take inﬁnitely many Lε-sentences as axioms of the Zermelo set theory. In the following, we shall use freely the abbreviations introduced in (2.27) as well as other abbreviations which will be introduced below.

9 (Extensionality) ∀x∀y (∀z (z ∈ x ↔ z ∈ y) → x ≡ y).(14) (Empty Set) ∃x∀y y ̸ε x.

In the following we use the expressions

(2.34) ∅ ε x, x ε ∅, zf-0 x ≡ ∅, ∅ ≡ x as abbreviations of

(15) (2.35) ∃y (∀z z ̸ε y ∧ y ε x), ∃y (∀z z ̸ε y ∧ x ε y), zf-1 ∃y (∀z z ̸ε y ∧ x ≡ y), ∃y (∀z z ̸ε y ∧ y ≡ x) respectively. By the Axiom of Extensionality, these formulas are equivalent to

(2.36) ∃!y (∀z z ̸ε y ∧ y ε x), ∃!y (∀z z ̸ε y ∧ x ε y), zf-2 ∃!y (∀z z ̸ε y ∧ x ≡ y), ∃!y (∀z z ̸ε y ∧ y ≡ x), respectively(16) . By the Axiom of Empty-set, these formulas are equivalent to

(2.37) ∀y (∀z z ̸ε y → y ε x), ∀y (∀z z ̸ε y → x ε y), zf-3 ∀y (∀z z ̸ε y → x ≡ y), ∀y (∀z z ̸ε y → y ≡ x), respectively.

(Pairing Axiom) ∀x∀y∃z∀u (u ε z ↔ (u ≡ x ∨ u ≡ y)).

In the following we use the abbreviations

(2.38) z ε {x, y}, {x, y} ε z, zf-4 {x, y} ≡ z, z ≡ {x, y} to denote the formulas

(2.39) ∃u (∀v (vεu ↔ (v ≡ x∨v ≡ y) ∧ z ε u), ∃u (∀v (vεu ↔ (v ≡ x∨v ≡ y) ∧ u ε z), zf-5

(14) In the next subsection, when we have introduced the system K∗ of logical deduction, we can show that ∀x∀y (x ≡ y → ∀z (z ∈ x ↔ z ∈ y)) is a provable sentence. Actually this formula is almost identical with an instance of one of the axiom schemas which guarantee that ≡ is interpreted as the equality. It follows that ∀x∀y (∀z (z ∈ x ↔ z ∈ y) ↔ x ≡ y) is a theorem provable form the Axiom of Extensionality.

(15) In the next subsection, when we have introduced the system K∗ of logical deduction, we can prove that this formula is equivalent to the formula “x 6≡ x” in K∗.

(16) What we really mean here is that, for example, the formula (∃y (∀z z 6ε y ∧ y ε x) ↔ ∃!y (∀z z 6ε y ∧ y ε x)) can be proved in the deduction system K∗ which is going to be introduced in the next subsection, under the Axiom of Extensionality. Similar remarks apply when we talk about implication, equivalence, or provability in the following.

10 ∃u (∀v (vεu ↔ (v ≡ x∨v ≡ y) ∧ z ≡ u), ∃u (∀v (vεu ↔ (v ≡ x∨v ≡ y) ∧ u ≡ z), respectively. Similarly to the case of “∅”, by the Axiom of Exensionality, we can replace the outermost quantiﬁcation ∃u in the formulas above with ∃!u. By the Pairing Axiom, we can prove the equivalence (in the system K∗ introduced below) of the formulas in (2.39) with the formulas

(2.40) ∀u (∀v (vεu ↔ (v ≡ x ∨ v ≡ y) → z ε u), ∀u (∀v (vεu↔(v ≡ x ∨ v ≡ y) → u ε z), zf-6 ∀u (∀v (vεu ↔ (v ≡ x∨v ≡ y) → z ≡ u), ∀u (∀v (vεu↔(v ≡ x∨v ≡ y) → u ≡ z), respectively. As in (1.2), we introduce the ordered pair ⟨x, y⟩ as the abbreviation of

′ (1.2) {{x}, {x, y}} where “{x}” is used as an abbreviation of “{x, x}”.

(Axiom of Union) ∀x∃y∀z (z ε y ↔ ∃u (u ε x ∧ z ε u)).

In connection with the Axiom of Union, we introduce the abbreviating notation of the union: S S (2.41) y ε x, x ε y, zf-7 S S y ≡ x, x ≡ y are abbreviations of

(17) (2.42) ∃z (∀u (u ε z ↔ ∃v (v ε x ∧ u ε v)) ∧ y ε x), zf-8 ∃z (∀u (u ε z ↔ ∃v (v ε x ∧ u ε v)) ∧ z ε y), ∃z (∀u (u ε z ↔ ∃v (v ε x ∧ u ε v)) ∧ y ≡ z), ∃z (∀u (u ε z ↔ ∃v (v ε x ∧ u ε v)) ∧ y ≡ z) respectively. The same kind of remarks about necessary renaming of the bounded variables in the formulas similar to the remarks after (2.35) or (2.39) and remarks on equivalent formulas like the ones in (2.36) and (2.37) also apply here. The formulation of the Axiom of Separation in the na¨ıve axiomatic setting was am- biguous in that we could not specify exactly what the ”property” was. We can now introduce this axiom without any ambiguity as a scheme of axioms in the following way

For each Lε-formula φ = φ(x, x0, ..., xn−1) we take the formula of the following form as an instance of the axiom scheme of the Axiom of Separation:

(Axiom of Separation)φ ∀x0 · · · ∀xn−1∀y∃u∀x (x ε u ↔ (x ε y ∧ φ(x, x0, ..., xn−1))).

(17) By the Axiom of Union this formula will be proved (in K∗ to be equivalent with ∃z (z ε x ∧ y ε z))

11 S (Axiom of Inﬁnity) ∃x (∅ ε x ∧ ∀y (y ε x → {y, {y}})).

(Power-set Axiom) ∀x∃y∀u (u ε y ↔ u ⊆ x).

Here “u ⊆ x” is an abbreviation of “∀v (v ε u → v ε x)”. The axioms introduced so far make up the axiom system of Zermelo set theory (denoted by Z) the axiom system contains inﬁnitely many sentences because of the Axiom (Scheme) of Separation.

The Axiom of Choice can be now formulated as an Lε-sentence as follows:

(Axiom of Choice) ∀x (∅ ̸ε x → ∃y (∀u (u ε y → ∃v∃w (v ε x ∧ (w ε x ∧ ⟨v, w⟩ ≡ u)) ∧ ∀v (v ε x → ∃!w (⟨v, w⟩ ε y))))) Note that the subformulas of the formula of the Axiom of Choice express that “y is a choice function on x”:

∀ ∅ ̸ → ∃ ∀ → ∃ ∃ ∧ ∧ ⟨ ⟩ ≡ (2.43) x ( ε x y ( |u (u ε y v w (v ε x{z (w ε v v, w u)))} zf-9 y consists of elements of the form ⟨v,w⟩ such that v∈x and w∈v ∧ ∀ → ∃ ⟨ ⟩ |v (v ε x {z!w ( v, w ε y))})). y is a function on x The axiom system consisting of axioms of Z and the Axiom of Choice is denoted by ZC. The axiom system of Zermelo-Fraenkel set theory (notation: ZF) consists of axioms of Z and the following two kinds of axioms. The ﬁrst one, Axiom of Replacement is again an axiom scheme and consists of inﬁnitely many Lε-sentences.

For each Lε-formula φ = φ(y, z, x0, ..., xn−1), let

(Axiom of Replacement)φ

∀x0 · · · ∀xn−1∀x (∀y (y ε x → ∃!z φ(y, z, x0, ..., xn−1))

→ ∃u∀v (v ε u ↔ ∃y (y ε x ∧ φ(y, v, x0, ..., xn−1)))).

(Axiom of Regularity) ∀x (x ̸≡ ∅ → ∃y (y ε x ∧ ∀z (z ε x → ¬z ε y))).

The axiom system of Zermelo-Fraenkel set theory with Axiom of Choice (notation: ZFC) is the axiom system consisting of the axioms of ZF and the Axiom of Choice.

2.3 The deduction system K∗ ded-sys logic-10-2019-05- In the following we are going to introduce a deduction system for the predicate logic 15.pdf which we call K∗.(18)

(18) This is just a quite arbitrary name and has no special meaning in it except that the letter “K” is chosen because of the German expression “logisches Kalk¨ul”which means logical deduction system.

12 For this, we have to introduce ﬁrst the propositional logic and the notion of tautology. In the following we are still working in meta-mathematics.

Let A0, A1, A2 ,... be (a potentially inﬁnite collection of) symbols which we call propositional symbols. Formulas of the propositional logic are sequences of symbols de- ﬁned inductively by:

(2.44) Each sequence of length 1 consisting of a propositional symbol is a formula of prop-logic-0 the propositional logic;

(2.45) If φ and ψ are formulas of the propositional logic, then (φ → ψ), ¬ψ are formulas prop-logic-1 of the propositional logic;

(2.46) Nothing else. prop-logic-2

For example (¬((A0 → A1) → ¬A4) → A5) is a formula of the propositional logic.

For a formula φ of the propositional logic, we write φ = φ(A0, ..., An−1) if all the propositional symbols appearing in φ are among A0, ..., An−1. For example, if we denote the formula of the propositional logic given in the example above as φ, we have

φ = φ(A0,A1,A2,A3,A4,A5) but we can also write φ = φ(A0,A1,A4,A5). { } (19) n × · · · × {⟨ ⟩ ∈ } Let 2 = 0, 1 and (2) = |2 {z 2} = i0, ..., in−1 : i0, ..., in−1 2 for n n ∈ N.

For formulas φ = φ(A0, ..., An−1) of propositional logic, we deﬁne their functional n → interpretation fφ(A0,..., An−1) :(2) 2 recursively as follows:

(2.47) If φ is the sequence “A ” for some k < n then prop-logic-3 k ( 1, if i = 1; ⟨ ⟩ k fφ(A0,..., An−1)( i0, ..., in−1 ) = 0, otherwise n for all ⟨i0, ..., in−1⟩ ∈ (2) ; (20) (2.48) If φ = (φ → φ ), then prop-logic-4 0 1   1, if f (⟨i , ..., i − ⟩) = 0 or  φ0(A0,..., An−1) 0 n 1 ⟨ ⟩ f (⟨i , ..., i − ⟩) = fφ1(A0,..., An−1)( i0, ..., in−1 ) = 1; φ(A0,..., An−1) 0 n 1   0, otherwise n for all ⟨i0, ..., in−1⟩ ∈ (2) ;

(2.49) If φ = ¬φ0, then ( prop-logic-5 1, if f (⟨i , ..., i − ⟩) = 0; ⟨ ⟩ φ0(A0,..., An−1) 0 n 1 fφ(A0,..., An−1)( i0, ..., in−1 ) = 0, otherwise n for all ⟨i0, ..., in−1⟩ ∈ (2) .

(19) The intended interpretation of 0 and 1 are “false” and “true”.

(20) Note that we have φ0 = φ0(A0, ..., An−1) and φ1 = φ1(A0, ..., An−1).

13 A formula φ = φ(A0, ..., An−1) in propositional logic is a tautology if fφ(A0,..., An−1) takes always the value 1. This is well deﬁned: we can prove easily that whether φ is tautology or not does not depend on the choice of the list of variables A0, ..., An−1. For a language L, L-formula ψ (in predicate logic) is said to be a tautology if there is a tautology φ = φ(A0, ..., An−1) and L-formulas φ0, ..., φn−1 such that ψ is obtained from

φ by replacing A0, ..., An−1 by φ0, ..., φn−1 respectively. By the deﬁnition of functional interpretation of formulas of propositional logic given by (2.47) through (2.49), tautologies in predicate logic should be formulas which are true in any interpretation as long as the logical connectives ”→” and ”¬” are interpreted as logical “then” and “not”. For each language L, we introduce the notion of the deduction in the system K∗ of predicate logic by introducing Axioms and Deduction Rules and deﬁning proofs in K∗ (of an L-formula form a set T of L-formulas).

In the following, we write φ0 → · · · φk−1 → φ for formulas φ0, ..., φk−1, φ to denote

(2.50) (φ0 → (φ1 → ( ··· (φk−2 → (φk−1 → φ))···)) pred-logic-5-0

As a special case of this notation, we often drop the parentheses in (φ → ψ) and write φ → ψ as far as no confusion occurs.

Axioms of K∗ in L:

• All L-formulas which are tautologies.

• (Axioms of Equality) All L-formulas of the following form

(2.51) x ≡ x; pred-logic-6

(2.52) x ≡ y → y ≡ x; pred-logic-7

(2.53) x ≡ y → y ≡ z → x ≡ z; pred-logic-8

(2.54) u0 ≡ v0 → · · · → um−1 ≡ vm−1 → f(u0, ..., um−1) ≡ f(v0, ..., vm−1) pred-logic-9 where f is an m-ary function symbol in L;

(2.55) u0 ≡ v0 → u1 ≡ v1 → · · · → un−1 ≡ vn−1 → r(u0, ..., un−1) → r(v0, ..., vn−1) pred-logic-10 where r is an n-ary relation symbol in L.

• (Axiom of Substitution) All L-formulas of the form

(2.56) φ(t/x) → ∃xφ pred-logic-11

where t is substitutable for x in φ.(21)

(21) φ(t/x) denotes the formula obtained form φ by substituting t for all free appearance of x in φ. t is substitutable for x in φ if all free appearances of x are not in the scope of the quantiﬁcation of any of the variables appearing in t.

14 Deduction rules of K∗ in L: All diagrams with L-formulas of the following form:

• (Modus Ponens) φ, φ → ψ ψ • (Rule of Existential Quantiﬁcation) φ → ψ ∃xφ → ψ where x does not appear in ψ as a free variable.

For a concretely given collection Γ of L-formulas and an L-formula φ, a proof of φ ∗ from Γ (in K ) is a ﬁnite sequence φ0, ..., φn such that

∗ (2.57) each φi, i ≤ n is either an Axiom of K , or one of the formulas in Γ, or there are pred-logic-12 φ , φ i , i < i such that i0 i1 is an instance of Modus Ponens, or there is i < i 0 1 φ 0 φ 1 such that i0 is an instance of the Rule of Existential Quantiﬁcation; and φi (2.58) φn = φ. pred-logic-13

∗ If P is a proof of φ from Γ, we write Γ ⊢K ,P φ. If there is some proof P of φ from ∗ Γ, we write Γ ⊢K φ. Some times we drop the superscript K∗ and write simply Γ ⊢P φ or Γ ⊢ φ. ∗ ∗ ∗ ∗ If Γ is empty set, we write ⊢K ,P φ, ⊢K φ, ⊢P φ, ⊢ φ instead of Γ ⊢K ,P φ,Γ ⊢K φ, Γ ⊢P φ,Γ ⊢ φ respectively. bbd-all: p.37- L-pred-logic-0

Lemma 2.1 For a collection Γ of L-formulas and L-formulas φ0, ..., φk−1, φ, if

(2.59) Γ ⊢ φ0 → φ1 → · · · → φk−1 → φ and pred-logic-14

(2.60) Γ ⊢ φ0, Γ ⊢ φ1,..., Γ ⊢ φk−1, pred-logic-15 then Γ ⊢ φ.

Proof. Let P be a proof of (2.59) and P0,..., Pk−1 the proofs of (2.60). Then

⌢ ⌢ ⌢ ⌢ ⌢ ⌢ (2.61) P P0 ··· Pk−1 ⟨φ0 → · · · → φk−1 → φ⟩ ⟨φ1 → · · · → φk−1 → φ⟩ ··· ⌢ ⌢ ⟨φk−1 → φ⟩ ⟨φ⟩ is a proof of φ from Γ. Note that, for example, φ1 → · · · → φk−1 → φ is obtained from the last formulas of P and P0 by Modus Ponens. (Lemma 2.1) In the following, we shall often cite Lemma 2.1 simply as Modus Ponens. L-pred-logic-0-a Lemma 2.2 ⊢ ∃x x ≡ x.

Proof. By the Axiom of Substitution, we have ⊢ x ≡ x → ∃x x ≡ x. By the Axiom of Equality (2.51), we have ⊢ x ≡ x. Hence, by Modus Ponens (Lemma 2.1), it follows that

⊢ ∃x x ≡ x. (Lemma 2.2)

15 Theorem 2.3 (Deduction Theorem) ( 1 ) For any collection of L-formulas Γ and L- T-pred-logic-0 fromulas φ, ψ, if Γ ⊢ φ → ψ then Γ, φ ⊢ ψ.(22) ( 2 ) For Γ and φ, ψ as in ( 1 ), if φ is a L-sentence then Γ ⊢ φ → ψ ⇔ Γ, φ ⊢ ψ.

Proof. ( 1 ): If Γ ⊢ φ → ψ, then we have Γ, φ ⊢ φ → ψ. We also have Γ, φ ⊢ φ (Note that ⟨φ⟩ is a proof(23) ). Thus, by Lemma 2.1, we have Γ, φ ⊢ ψ. ( 2 ): By ( 1 ), it is enough to show “⇒”. We show

P Q (2.62) If Γ, φ ⊢ ψ, then, starting from P, we can construct a proof Q with Γ ⊢ φ → ψ pred-logic-16 by induction on the number n of formulas in P (that is, by the length of P as a sequence). If n = 1, then ψ is either ( i ) a logical axiom or ( ii ) a formula in Γ or (iii) φ. If ( i ) or ( ii ) holds then, since ψ → φ → ψ is a tautology, the sequence ⟨ψ, ψ → φ → ψ, φ → ψ⟩ is a proof of φ → ψ from Γ. If (iii) holds then φ → ψ is identical with ψ → ψ which is a tautology. Thus ⟨φ → ψ⟩ is a proof of φ → ψ form Γ. Now assume that (2.62) holds for all proofs with less than n formulas and assume that Γ, φ ⊢P ψ where P as a sequence has the length n. Case I. ψ is either a logical axiom or one of the formulas among Γ, φ. In this case, we can prove φ → ψ from Γ just as in the case of n = 1. Case II. ψ is deduced by Modus Ponens from some formulas in P. Then there is some L-fromula η such that η and η → ψ appear in P and ψ is deduced from these two formulas. The initial segments of P to each of these formulas are shorter than P hence by indcuction P P hypothesis there are proofs P0, P1 such that Γ ⊢ 0 φ → η and Γ ⊢ 1 φ → (η → ψ). Since (φ → η) → (φ → (η → ψ)) → (φ → ψ) is a tautology, denoting this formula θ, ⌢ ⌢ and the formula (η → ψ)) → (φ → ψ), the sequence P0 P1 ⟨θ, θ0, φ → ψ⟩ is a proof of φ → ψ by Lemma 2.1. Case III. ψ is of the form ∃x η → ξ where x does not appear in ξ as free variable and η → ξ appears in P. Let P′ be the initial segment of P ending with η → ξ. ′ Since Γ, φ ⊢P η → ξ and P′ is shorter than P, we have Γ ⊢ φ → η → ξ. Since (φ → η → ξ) → (η → φ → ξ) is a tautology, we get Γ ⊢ η → φ → ξ. Remembering that φ is a sentence, we may apply the Rule of Existential Quantiﬁcation to obtain Γ ⊢ ∃x η → φ → ξ. Now since (∃xη → φ → ξ) → (φ → ∃xη → ξ) is a tautology, we get

Γ ⊢ φ → ∃x η → ξ. (Theorem 2.3) | {z } further “proof theory”: logic-12- =ψ 2019-05-22.pdf A (concretely given) collection of L-sentences T is called a(n L-)theory. L-pred-logic-0-0 Lemma 2.4 For any language L, L-theory T and L-formulas φ, ψ, T ⊢ φ and T ⊢ ψ ⇔ T ⊢ (φ ∧ ψ).

(22) For a collection Γ of L-fromulas and L-formulas φ, ψ, we simply write Γ, φ ` ψ in place of Γ∪{φ} ` ψ.

(23) In abuse of notation, we denote with ha0, ..., an−1i the sequence whose ith component is ai for i < n.

Strictly speaking, we have to write {h0, a0i} instead of ha0i and {h0, a0i, h1, a1i} instead of ha0, a1i etc.

16 Proof. Suppose that T ⊢ φ and T ⊢ ψ. Since φ → ψ → (φ ∧ ψ) is a tautology, we have T ⊢ φ → ψ → (φ ∧ ψ). If follows from Lemma 2.1 that T ⊢ (φ ∧ ψ). Suppose now that T ⊢ (φ ∧ ψ). Since (φ ∧ ψ) → φ and (φ ∧ ψ) → ψ are tautologies, we have T ⊢ (φ ∧ ψ) → φ and T ⊢ (φ ∧ ψ) → ψ. By Lemma 2.1, it follows that T ⊢ φ and T ⊢ ψ. (Lemma 2.4)

An L-theory T is inconsistent if T proves any L-formula. Otherwise T is consistent.

L-pred-logic-1 Lemma 2.5 For any language L and L-theory T the following are equivalent: ( a ) T is inconsistent; ( b ) For some L-formula φ, we have T ⊢ φ and T ⊢ ¬φ; ( c ) T ⊢ (φ ∧ ¬φ) for some L-formula φ; ( d ) T ⊢ x ̸≡ x.

Proof. ( a ) ⇒ ( b ): Clear by deﬁnition. ( b ) ⇔ ( c ): By Lemma 2.4. ( c ) ⇒ ( d ): Asuume that T ⊢ (φ∧¬ψ) for some L-formula φ. Since (φ∧¬φ) → x ̸≡ x is a tautology, we have T ⊢ (φ ∧ ¬φ) → x ̸≡ x. By Lemma 2.1 it follows that T ⊢ x ̸≡ x.

( d ) ⇒ ( a ): Assume T ⊢ x ̸≡ x. Since x ≡ x is one of the Axioms of Equality, we have T ⊢ x ≡ x. Hence, by Lemma 2.4, it follows that T ⊢ (x ≡ x ∧ x ̸≡ x). For any L-formula φ,(x ≡ x ∧ x ̸≡ x) → φ is a tautology. Thus T ⊢ (x ≡ x ∧ x ̸≡ x) → φ and

T ⊢ φ by Lemma 2.1. (Lemma 2.5)

L-pred-logic-2 Lemma 2.6 For any language L, collection of L-formula T , L-fromula φ and variables v0, ..., vn−1, we have

T ⊢ φ ⇔ T ⊢ ∀v0 · · · ∀vn−1φ

Proof. It is enough to prove the asserion for n = 1. Recall that ∀v x is an abbreviation of ¬∃v¬φ. (⇒): Suppose that T ⊢ φ and let ϕ be an arbitrary L-sentence which is a tautology. Since T, ψ ⊢ φ, we have T ⊢ ψ → φ by the Deduction Theorem (Theorem 2.3). Since (ψ → φ) → (¬φ → ¬ψ) is a tautology, we obtain T ⊢ ¬φ → ¬ψ by Lemma 2.1. Thus, by the Rule of Existential Quantiﬁcation, it follows that

(24) (2.63) T ⊢ ∃v¬φ → ¬ψ. pred-logic-17

Now, since (∃v¬φ → ¬ψ) → (ψ → ¬∃v¬φ) is a tautology, we obtain

(2.64) T ⊢ ψ → ¬∃v¬φ pred-logic-18

(24) Remember that ψ is a sentence and hence does not contain x as a free variable!

17 by Lemma 2.1. Since ψ is a tautology, we have T ⊢ ψ. Thus, again by Lemma 2.1, T ⊢ ¬∃v¬φ, that is, T ⊢ ∀v φ.

(⇐): This can be proved similarly to “⇒”. (Exercise). (Lemma 2.6)

Lemma 2.7 (Substitution Theorem) Suppose that L is a language and T a collection L-pred-logic-3 of L-formulas. If T ⊢ φ for an L-fromula and t is a L-term substitutable in the variable x in φ, Then T ⊢ φ(t/x).

Proof. By T ⊢ ¬φ → ¬φ. By the Axiom of Substitution, we have T ⊢ ¬φ(t/x) → ∃x¬φ. Since (¬φ(t/x) → ∃¬φ) → (¬∃x¬φ → φ(t/x)) is a tautology,

(2.65) ⊢ ¬∃x¬φ → φ(t/x) pred-logic-19 by Lemma 2.1. Since T ⊢ φ, we have

(2.66) T ⊢ ¬∃x¬φ pred-logic-20 by Lemma 2.6. Thus, by (2.65), (2.66) and Lemma 2.1, we obtain T ⊢ φ(t/x). (Lemma 2.7)

2.4 Predicate logic in Z and the Completeness Theorem comp katowice-03-2019- 2.5 Some semantic proofs of proof theoretic assertions 10-14

The following are applications of Soundness Theorem and Completeness Theorem. Both of the following theorems can be proved proof-theoretically (i.e., in meta-mathematically without appealing to any amount of set-theory). The proof-theoretical proofs are much more difficult than the ones presented here. For example, a possible proof of Theorem 2.8 is that we first establish the equivalence of our system K∗ to a Gentzen style deduction system and applying the Cut Elimination Theorem of Gentzen. The model theoretic proofs via Soundness and Completeness Theorems(25) are just a sort of first approxima- tions to the corresponding proof-theoretic proofs which can be much more complicated. !!!! T-pred-logic-1 Theorem 2.8 Suppose that T is an L-theory, φ an L-fromula, and L′ is a language extending L. Then we have

′ T ⊢L φ if and only if T ⊢L φ(26)

′ Proof. If T ⊢L φ then we clearly have T ⊢L φ. Suppose T ̸⊢L φ. Then, by Completeness Theorem, there is an L-structure A such that A |= T but A ̸|= φ.

(25) Such proofs are also often called semantical proofs in contrast to the syntactical or proof-theoretical proofs which can be done in the meta-mathematical frame-work without utilizing the notion of interpretation of formulas in models, and Soundness and Completeness Theorems.

(26) We denote with T `L φ the statement “there is a proof P of φ from T such that all the components of P are L-formulas.”

18 Let A˜ be an arbitrary expansion of A to a L′-structure. We still have A˜ |= T and L′ A˜ ̸|= φ. Thus we have T ̸⊢ φ by Soundness Theorem. (Lemma 2.8)

For a language L and L-theory T , an L′-theory T˜ with T˜ ⊇ T for a language L′ ⊇ L is said to be a conservative extension of T if, for any L-sentence φ, T˜ ⊢ φ implies T ⊢ φ.

T-pred-logic-2 Theorem 2.9 Suppose that T is an L-theory. ( 1 ) If

(2.67) T ⊢ ∃!xφ pred-logic-21 for an L-fromula φ = φ(x), then the theory T˜ = T ∪ {φ(c)} is a conservative extension of T where c is a new constant symbol not appearing in T ∪ {φ}. ( 2 ) If

(2.68) T ⊢ ∀x0 · · · ∀xk−1∃!x φ pred-logic-22 for an L-formula φ = φ(x, x0, ..., xk−1), then the theory T ∪ {∀x0 · · · ∀xk−1φ(f (x0, ..., xk−1), x0, ..., xk−1)} is a conservative extension of T where f is a new k-aray function symbol not appearing in T ∪ {φ}.

( 3 ) For any L-formula φ = φ(x0, ..., xk−1), the theory T ∪ {∀x0 · · · ∀xk−1(r(x0, ..., xk−1f ) ↔ φ(x0, ..., xk−1))} is a conservative extension of T where r is a new k-ary relation symbol not appearing in T ∪ {φ}.

Proof. We prove ( 2 ). Other assertions of the theorem can be treated similarly. Let T and φ be as in ( 2 ) and let T˜ = T ∪ {∀x0 · · · ∀xk−1φ(f (x0, ..., xk−1), x0, ..., xk−1)}. Let ψ be an L-formula and suppose T˜ ⊢ ψ. We have to show that T ⊢ ψ. By the Completeness Theorem, it is enough to show that T |= ψ. Suppose that A = ⟨A, ...⟩ be an L-structure with A |= T . Let

A′ (2.69) f = {⟨⟨a0, ..., ak−1, b⟩⟩ : A |= φ(b, a0, ..., ak−1)}. pred-logic-23

A′ k → ′ ⟨ A′ ⟩ ′ | ˜ By (2.68), f : A A. Let A = |A,{z ...}, f . Then A = T . By Soundness Theorem, A ′ ′ it follows that A |= ψ. Hence A = A ↾ L |= ψ. (Theorem 2.9)

3 Incompleteness Theorems and Speed-up Theorems

incompl 3.1 Ehrenfeucht-Mycielski Speed-up Theorem ehrenfeucht-

The meta-mathematical objects, like symbols, (finite) sequences of symbols, (finite) se- mycielski logic-15-2019-06- quences of (finite) sequences of symbols, etc. correspond elements of Vω. 10.pdf To make the coding of languages and logic in set theory easier, we expand the language

Lε of set thoery with a new constant symbol ∅ and two function symbols {·, ·} and · ∪ ·.

19 Let L{} be the extension of Lε obtained by adding the constant symbol ∅ and the binary function symbols {·, ·}, ·∪·. Let Z{} be the L{}-theory obtained from Z by adding the axioms which claim that the expected interpretation of the new symbols is valid.

Thus Z{} is a conservative extension of Z and, e.g., ∀x (x ̸ε ∅) is one of the axioms of

Z{}. We introduced the theory Z0 as a concretely given subtheory of Z{} which is strong enough for coding the meta-mathematical objects and notions in it. In the following, T is a concretely given (consistent) theory in L{} which extends Z0. Usually what we have in mind is such a theory T that is compatible with ZF or ZFC but we do not assume this unless we mention it explicitly.

We can express “all” concrete hereditarily ﬁnite sets by closed L{}-terms. In particular, for any concretely given number n, we have a closed L{}-term n which corresponds to the number n in Z{}: For 0, 1, 2, 3, etc., we set 0, 1, 2, 3 etc. to be ∅,(∅ ∪ {∅, ∅}), ((∅ ∪ {∅, ∅}) ∪ {(∅ ∪ {∅, ∅}), (∅ ∪ {∅, ∅})}), etc. In general, we deﬁne n + 1 to be n ∪ {n, n}. By induction on n (in meta-mathematics), we can show that for any n ∈ N

(3.1) Z0 ⊢ “∅ is the minimal element of ω”, incompl-a

(3.2) Z0 ⊢ “n ε ω” and incompl-0

(3.3) Z0 ⊢ “n + 1 is the successor of n”. incompl-1

Recall that we assumed that symbols of meta-mathematics are elements of N2. The collection of meta-mathematical finite sequences of symbols is denoted by Seq. We identify the pair ⟨m, n⟩ with the sequence of length 1 with the entry ⟨m, n⟩ and regard N2 as a subset of Seq with this identification. ⌜s⌝: corner quote ω> 2 of s We assign a closed L{}-term ⌜s⌝ with Z0 ⊢ “⌜s⌝ ∈ (ω )” to each symbol or finite sequence of symbols s such that if s is a sequence of length ℓ and its kth symbol is (coded by) the ordered pair ⟨m, n⟩ then Z0 ⊢ “the length of ⌜s⌝ is ℓ” and Z0 ⊢ “the kth symbol of ⌜s⌝ is ⟨m, n⟩”. Similarly ⌜·⌝ is extended to finite sequences of finite sequences of symbols, finite sequences of finite sequences of finite sequences of symbols etc. We cite the following theorem on a characterization of recursiveness without a proof.

Theorem 3.1 (Representability) For any S ⊆ Seq, S is recursive (i.e. the validity of T-incompl-a

“s ∈ S” is computable) if and only if there is an L{}-formula ψ = ψ(x) such that for any sequence s of symbols,

(3.4) if s ∈ S then Z0 ⊢ ψ(⌜s⌝) and

(3.5) if s ̸∈ S then Z0 ⊢ ¬ψ(⌜s⌝).

チェックする: inc:req:int: defn:representable-fn in https://builds.openlogicproject.org/open-logic-debug.pdf The following theorem will be proved later as Theorem 3.13:

20 Theorem 3.2 (Diagonal Lemma) For an arbitrary formula ψ in a language L extending T-incompl-0

L{}, there is an L-formula σ such that

(3.6) free(σ) ⊆ free(ψ) \{x0}, and incompl-2

(3.7) Z0 ⊢ σ ↔ ψ(⌜σ⌝/x0). incompl-3

Tarski’s Undeﬁnability of the Truth is an easy application of the Diagonal Lemma:

Corollary 3.3 (Tarski, Undeﬁnability of the Truth) For any language L extending L{} T-incompl-0-0 and consistent theory T in L with T ⊇ Z0, there is no L-formula χ = χ(x) such that T ⊢ φ ↔ χ(⌜φ⌝) for all L-sentences φ.(27)

Proof. For any L-formula χ, letting ψ to be the formula ¬χ, there is a L{}-sentence

φ such that Z0 ⊢ φ ↔ ψ(⌜φ⌝) by Diagonal Lemma (Theorem 3.2). Thus, we have

T ⊢ φ ↔ ¬χ(⌜φ⌝). This shows that χ is not a deﬁnition of the truth for T . (Corollary 3.3)

Remember that a proof P in K∗ is a sequence of finite sequences of symbols. We define the length L(P) of a proof P to be the length of the finite sequence obtained by concatenating all the components of the sequence P.

For an L{}-theory T ⊇ Z0, and τ ∈ Th(T ), let

(3.8) WT (τ) = the smallest possible length L(P) of a proof P of τ. incompl-7

Theorem 3.4 (Ehrenfeucht and Mycielski Speed-up Theorem, [Ehrenfeucht-Mycielski 1971])

Suppose that T is a concretely given consistent theory in a ﬁnite language L extending E-M-thm

L{} with T ⊇ Z0 and φ0 is independent from T . Then, there is no recursive function S : N → N such that

≤ ∈ incompl-8 (3.9) WT (τ) S(WT ∪{φ0}(τ)) for all τ Th(T ).

In other words, for any recursive S : N → N, there is a τ ∈ Th(T ) such that

S(WT ∪{φ0}(τ)) < WT (τ).

Theorem 3.4 follows from the next two Lemmas. Note that, in connection with Theorem 3.1, F : N → N is recursive if and only if F as a graph F ⊆ N2 ⊆ Seq is recursive. L-incompl-0 (28) Lemma 3.5 If Th(T ∪ {¬φ0}) is undecidable , there is no recursive F : N → N such that

(3.10) WT (τ) ≤ F (WT (φ0 → τ)) for all τ ∈ Th(T ). incompl-9

Proof. Note that we have

21 K∗ K∗ K∗ (3.11) T, ¬φ0 ⊢ σ ⇔ T ⊢ ¬φ0 → σ ⇔ T ⊢ (φ0 ∨ σ) incompl-10 by the Deduction Theorem (Theorem 2.3) of K∗. Suppose, toward a contradiction, that there is a recursive function F : N → N satisfying (3.10). Without loss of generality, we may assume that F is increasing(29) . Now, by (3.10), we have

(3.12) WT (φ0 ∨ σ) ≤ F (WT (φ0 → (φ0 ∨ σ))) holds for any σ ∈ Th(T ). incompl-11

Note that σ ∈ Th(T ) implies that (φ0 ∨ σ) ∈ Th(T ).

Since (φ0 → (φ0 ∨ σ)) is a tautology, there is a recursive function (actually a linear function) G : N → N such that

(3.13) WT (φ0 → (φ0 ∨ σ)) = ℓn(φ0 → (φ0 ∨ σ)) = G(ℓn(σ)) incompl-12

Thus

∨ ≤ → ∨ ≤ (30) (3.14) WT (φ0 σ) | {z } F (WT (φ0 (φ0 σ))) F (G(ℓn(σ))). incompl-13 by (3.12)

This means that, for each L-sentence σ, we ﬁnd a proof of (φ0 ∨ σ) form T in the

ﬁnite collection of all the proofs in L of length ≤ F (G(ℓn(σ))), if (φ0 ∨ σ) has a proof (31) from T . By (3.11), this is a contradiction to the undecidability of Th(T ∪ {¬φ0}).

(Lemma 3.5) logic-2019- bbd-all.pdf, p.55 Lemma 3.6 There is a recursive function R : N → N such that L-incompl-1

(32) ≤ → ∈ ∪ { } L-incompl-2 (3.15) WT ∪{φ0}(τ) R(WT (φ0 τ)) holds for all τ Th(T φ0 ). Proof. The proof of the Deduction Theorem for K∗ gives an algorithm to obtain a proof of τ from T ∪ {φ0} from a given proof of φ → τ form T . (Lemma 3.6) Proof of Ehrenfeucht-Mycielski Speed-up Theorem 3.4: Suppose, toward a contradiction, that there is a recursive S : N → N satisfying (3.9). Without loss of generality, we may assume that S is increasing. We have

(3.16) W (τ) ≤ S(W ∪{ }(τ)) ≤ S(R(W (φ → τ))). incompl-14 T | {z } T φ0 | {z } T 0 by the assumption of (3.9) by Lemma 3.6

(27) Such χ (if it would exist) is called a deﬁnition of the truth for T .

(28) By Theorem 3.15, this condition is equivalent to the consistency of T ∪ {¬φ0}. (29) Otherwise we may replace F by n 7→ max{F (k): k ≤ n}

(30) For sequence s ℓn(s) denotes the length of s.

(31) Note that the ﬁniteness of the correction of proofs (modulo renaming of the variables) follows from the ﬁniteness of L.

(32) Note that by Deduction Theorem (Theorem 2.3), τ ∈ Th(T ∪ {φ0}) if and only if φ0 → τ ∈ Th(T ). ∈ ∪ { } → Thus, if τ Th(T φ0 ), that is, if WT ∪{φ0} is deﬁned at τ then, WT is deﬁned at φ0 τ.

22 Since the composition of R and S is recursive, this is a contradiction to Lemma 3.5. Note that T ∪ {¬φ0} is consistent and hence undecidable by Theorem 3.15. (Lemma 3.6) The Ehrenfeucht-Mycielski Theorem does not hold if we modify the notion of the length of a proof to be the number of formulas in the proof:

T-length-0 Theorem 3.7 For any concretely given theory T in an arbitrary language L, there is a recursive theory T ∗ in L such that

∗ (3.17) T and T are equivalent (i.e. they deduce the same sentences); three-0 ∗ (3.18) Each theorem of T (or equivalently, of T ) has a proof consisting of exactly three three-1 formulas.

Proof. Let ⟨ψn : n ∈ N⟩ be a recursive enumeration of all the theorems of T . For each n ∈ N, let

∧ · · · ∧ (33) (3.19) ηn = |ψn {z ψn}. three-2 n+1-times

Note that (ηn → ψn) is a tautology for all n ∈ N. In particular, all such formulas are axioms of the deduction system K∗. We show that

∗ (3.20) T = {ηn : n ∈ N} three-3 is as desired. ∗ T is recursive: Suppose that φ is an L-formula of length k if φ is one of η0,..., ηk−1 then φ is an axiom of T ∗. Otherwise, it is not. ∗ By the deﬁnition of T ∗, it is clear that T ∗ satisﬁes (3.17). If T ∗ ⊢K φ for an L- K∗ sentence φ, then T ⊢ φ. Hence, there is n ∈ N such that φ = ψn. Then the sequence ∗ ηn,(ηn → ψn), ψn is a proof of φ in T . (Theorem 3.7) The theorem above is a variation of Craig’s Theorem [Craig 1953]. With a similar proof, we can even prove the improvement of Theorem 3.7 in which T may be only recursively enumerable and T ∗ can be taken to be primitive recursive (Exercise).

3.2 Diagonal lemma diagonal logic-17-2019-07- In the following, we work (mainly) in meta-mathematics. 01.pdf

As before, Z denotes the Zermelo set theory in the language Lε = {ε}. Z{} is the canonical expansion of Z in L{} = {∅, {·, ·}, · ∪ ·, ε}. Z0 is the minimal subset of Z{} containing everything we need in the following discussion, and T , T ′ etc. are concretely given theories extending Z0.

(33) “n + 1” is set here to make η0 not an empty sequence but η0 itself.

23 In Z0, Vω is only a deﬁnable class, since the existence of Vω cannot be proved in Z

([Mathias 2001]). Note that the formula “x ε Vω” can be deﬁned by

(3.21) (∃n ε ω)(“x ε Vn”). diag-0

Note that, for any closed L{}-term t, we have Z0 ⊢ t ε Vω. Recall that we assumed that, in meta-mathematics, all the symbols are pairs of numbers, that is, elements of N2 and thus the collection of finite sequence Seq of symbols ω> 2 in meta-mathematics corresponds to the set (ω) in Z0. Note that, in contrast to Vω, ω> 2 ω> 2 (ω) is a set (i.e. Z0 ⊢ ∃x(“x ≡ (ω )”) ). In meta-mathematics, we also have to consider finite sequences of finite sequences of symbols. Proofs, for example, are such objects. We denote the collection of all finite sequences of finite sequences of symbols by Seq(Seq).

For a ﬁnite sequence s ∈ Seq of symbols, we deﬁned a closed L{}-term ⌜s⌝ which represents the set in Z0 which corresponds to s. If s is a sequence of length ℓ and its kth symbol is (coded by) the ordered pair ⟨m, n⟩ then Z0 ⊢ “the length of ⌜s⌝ is ℓ” and

Z0 ⊢ “the kth symbol of ⌜s⌝ is ⟨m, n⟩”. For s ∈ Seq(Seq), ⌜s⌝ is deﬁned similarly. L In Z0, TermL{} denotes the set of all {}-terms. That is, TermL{} is the set in Z0 which corresponds to the collection of L{}-terms in meta-mathematics. We have Z0 ⊢ ⊆ ω> 2 “TermL{} (ω )”. Hence, TermL{} is really a set in the theory Z0 by the instance of the

Axiom of Separation (which should be) available in Z0. Similarly, let ClTermL{} , FmlL{} , L L L and SentL{} be the set of all closed {}-terms, all {}-formulas, and all {}-sentences, respectively. We do not introduce these as new constant symbols but rather treat them ∈ as auxiliary symbols. For example, if we write “t TermL{} ”, we consider this as an abbreviation of an L{}-formula which expresses that “t is an L{}-term”. More generally, for an arbitrary language L, that is, for an arbitrary subset L of

{4, 5, 6} × ω extending L{}, we let TermL, ClTermL, FmlL, and SentL be the set of all L-terms, all closed L-terms, all L-formulas, and all L-sentences, respectively.

For each concretely given number n, we can deﬁne the closed L{}-term n which is called a numeral of n by

(3.22) 0 = ∅; and diag-1 n + 1 = n ∪ {n, n} for all n ∈ N.

· Vω → Vω In Z0, we can deﬁne a class function ( ) : ClTermL{} Vω such that (t) is the interpretation of the closed L{}-term t in the structure ⟨Vω, ∅, {·, ·}, · ∪ ·, ∈⟩. The formula “(x)Vω ≡ y” can be deﬁned as

⊆ (3.23) “x ε ClTermL{} and there are z and f such that z ClTermL{} x ε z, z is closed diag-1-a with respect to sub-terms, f is a mapping on z, f(“∅”) is the empty set(34) ,

(34) “∅” denotes here the sequence of symbols of length 1 whose single entry is the symbol ‘ ∅ ’.

24 if “{t0, t1}” ε z then f(“{t0, t1}”) is the set consisting of f(t0) and f(t1), if

“t0 ∪ t1” ε z then f(“t0 ∪ t1”) is the union of f(t0) and f(t1), and y ≡ f(x). L-diag-0

Lemma 3.8 ( 1 ) For every closed L{}-term t, we have Z0 ⊢ t ε Vω.

⊢ ∀ Vω ( 2 ) Z0 “( t ε ClTermL{} )((t) ε Vω)”.

⊢ ∀ ∃ Vω ≡ ( 3 ) Z0 “( x ε Vω)( t ε ClTermL{} )((t) x)”.

ω> 2 In Z0, let ⊏ be a ﬁxed deﬁnable well-ordering on (ω ) of order type ω. For all ∈ ⊏ ∈ Vω a Vω, there is the -minimal t ClTermL{} such that (t) = a by Lemma 3.8, ( 3 ). We ⌞ ⌟ ⌞·⌟ → denote this t by a . Actually, we introduced here a class function : Vω ClTermL{} . We have the following:

L-diag-1 Vω Lemma 3.9 ( 1 ) Z0 ⊢ (∀x ε Vω) ((⌞x⌟) ≡ x).

⊢ ∀ Vω ≡ ⌞ Vω ⌟ Vω ( 2 ) Z0 ( t ε ClTermL{} ) ((t) ( (t) ) ).

For a concretely given sequence s of symbols, let ⌜s⌝ be the closed L{}-term which expresses the element of ω>(ω2) corresponding to s, we choose ⌜s⌝ to be the ⊏-minimal (35) among such closed L{}-terms where ⊏ here is the binary relation on sequences on symbols in meta-mathematics which corresponds to the relation ⊏ in Z0. Since ⊏ in Z0 is definable, we can use the same definition in the meta-mathematics to introduce the meta-mathematical version of ⊏. Also we modify the definition of the numerals n, n ∈ N such that n is the ⊏-minimal L ⊢ ≡ closed {}-term t with Z0 n0 t where n0 denotes the original numeral for n as defined in (3.22). By (3.22) and the modified definition above, we have

(3.24) Z0 ⊢ 0 ≡ ∅; and diag-1-0

Z0 ⊢ n + 1 ≡ n ∪ {n, n} for all n ∈ N. L-diag-2 Lemma 3.10 For any concretely given closed L{}-term t and number n ∈ N, we have

( 1 ) Z0 ⊢ ⌞t⌟ ≡ ⌜t⌝,

( 2 ) Z0 ⊢ ⌞n⌟ ≡ ⌜n⌝,

Vω ( 3 ) Z0 ⊢ (⌜t⌝) ≡ t, and

Vω ( 4 ) Z0 ⊢ (⌜n⌝) ≡ n.

Note that, in ( 1 ) above, ⌞·⌟ is an operation in Z0 while ⌜·⌝ is an “operation” in met- mathematics, and ⌜t⌝ for a concretely given L{}-term t is another concretely given L{}- term.

Since T is concretely given, we can introduce a predicate “φ ε ⌜⌜T⌝⌝” in L{} which corresponds the meta-mathematical “φ is an L{}-sentence in T ”. Since {φ ε SentL : φ ε

(35) Note that, for example, L{}-terms “{∅, {∅, ∅}}”, “{{∅, ∅}, ∅}”, “{{∅, ∅}, {∅, ∅}} ∪ {∅, ∅}” are proved to be ≡ to each other in Z0.

25 ⌜⌜T⌝⌝} ⊆ ω>(ω2) is a set, we treat the symbol ⌜⌜T⌝⌝ also to denote this set though we do not introduce this symbol formally as a new constant symbol. ⌜⌜T⌝⌝ should be a representation of T in the following sense:

L-diag-3

Lemma 3.11 For any L-sentence φ, ( 1 ) if φ is in T , then, we have Z0 ⊢ ⌜φ⌝ ε ⌜⌜T⌝⌝, and, ( 2 ) if φ is not in T , then we have Z0 ⊢ ⌜φ⌝ ̸ε ⌜⌜T⌝⌝. × × → In Z0, let Subst : FmlL ω TermL{} FmlL be the deﬁnable function which is introduced by

(3.25) Subst(φ, n, t) ≡ “ the ψ ε FmlL obtained by replacing all the free appearance diag-2

of the variable xn (= ⟨0, n⟩) in φ with the term t ”. L-diag-4 Lemma 3.12 Suppose that φ is a concretely given L-formula, s a concretely given sequence of symbols, and k ∈ N.

( 1 ) Z0 ⊢ ⌞⌜s⌝⌟ ≡ ⌜(⌜s⌝)⌝.

( 2 ) If s is an L{}-term, then Z0 ⊢ Subst(⌜φ⌝, n, ⌜s⌝) ≡ ⌜φ(s/xn)⌝.

( 3 ) Z0 ⊢ Subst(⌜φ⌝, n, ⌜k⌝) ≡ ⌜φ(k/xn)⌝

( 4 ) Z0 ⊢ Subst(⌜φ⌝, n, ⌞⌜s⌝⌟) ≡ ⌜φ(⌜s⌝/xn)⌝.

( 5 ) Z0 ⊢ Subst(⌜φ⌝, n, ⌞k⌟) ≡ ⌜φ(k/xn)⌝.

Proof. ( 1 ): By Lemma 3.10, ( 1 ). ⊢ ⌜ ⌝ ⊢ ⌜ ⌝ ⌜ ⌝ ≡ ( 2 ): Note that we have Z0 s ε TermL{} . This as well as Z0 Subst( φ , n, s )

⌜φ(s/xn)⌝ can be proved by induction on the deﬁnition of ⌜s⌝ along with the construction ′ of s — for the induction step, use: φ(f(t0, ..., tℓ)/xn) = φ (t0/v0, ..., tℓ−1/vℓ−1). ( 3 ): By ( 2 ). ( 4 ): By ( 1 ) and ( 2 ).

( 5 ): By ( 3 ) and Lemma 3.10, ( 2 ). (Lemma 3.12)

The following Theorem 3.13, which we already cited as Theorem 3.2, is also known as the Fixed-Point Theorem. This theorem is the essence of the proofs of Gödel’sIncom- pleteness Theorems in [Gödel1932]. Theorem 3.13 was extracted from the original proofs in [Gödel1932] of these theorems by R. Carnap in [Carnap 1934a] and [Carnap 1934b].

Theorem 3.13 (Diagonal Lemma) For an arbitrary formula ψ in a language L extend- diagonal-lemma ing L{}, there is an L-formula σ such that

(3.26) free(σ) ⊆ free(ψ) \{x0}, and diag-2-0

(3.27) Z0 ⊢ σ ↔ ψ(⌜σ⌝/x0). diag-2-1

Proof. If x0 is not free in ψ, we can simply take σ to be ψ. Thus, we may assume that x ∈ free(ψ). 0 ∗ ω> 2 2 ω> 2 In Z0, let f : (ω ) → (ω ) be deﬁned by

26 ( ∗ u if s ε FmlL and u ≡ Sbst(s, 0, ⌞t⌟) (3.28) f (s, t) = diag-3 ∅ otherwise.

Cl-diag-0 Claim 3.13.1 Suppose that φ is a concretely given L-formula and s a concretely given ∗ sequence of symbols. Then Z0 ⊢ f (⌜φ⌝, ⌜s⌝) ≡ ⌜φ(⌜s⌝/x0)⌝.

∗ ⊢ By Lemma 3.12, ( 1 ) and (3.28), Z0 ⊢ f (⌜φ⌝, ⌜s⌝) ≡ Subst(⌜φ⌝, 0, ⌜⌜s⌝⌝). By Lemma 3.12, ( 2 ), ∗ it follows that Z0 ⊢ f (⌜φ⌝, ⌜s⌝) ≡ ⌜φ(⌜s⌝/x0)⌝. ⊣ (Claim 3.13.1)

In meta-mathematics, let k be the ﬁrst index such that xk does not appear in ψ. Let ∗ ∗ (3.29) s be the closed L{}-term ⌜∀xk (f (x0, x0) ≡ xk → ψ(xk/x0))⌝, and let diag-4 ∗ ∗ ∗ (3.30) σ be the L-sentence ∀xk (f (s , s ) ≡ xk → ψ(xk/x0)). diag-5 We show that this σ is as desired. Clearly σ satisﬁes (3.26). By (3.29) and Claim 3.13.1, we have

⊢ ∗ ∗ ∗ ≡ ⌜∀ ∗ ∗ ∗ ≡ → ⌝ (3.31) Z0 f (s , s ) |xk (f (s , s ) {zxk ψ(xk/x0))} . diag-6 =⌜σ⌝ by (3.30) Thus, by (3.30) and (3.31),

(3.32) Z0 ⊢ (σ → ψ(⌜σ⌝/x0)). diag-7 On the other hand, noting that

∗ ∗ ∗ (3.33) (ψ(⌜σ⌝/x0) → (f (s , s ) ≡ xk → ψ(⌜σ⌝/x0))) diag-8 is a tautology, and that xk does not appear in ψ by the choice of k, we have ∗ ∗ ∗ (3.34) Z0 ⊢ (ψ(⌜σ⌝/x0) → ∀xk (f (s , s ) ≡ xk → ψ(⌜σ⌝/x0))) diag-9 by Axiom of Substitution and Modus Ponens. By (3.31), it follows that ⊢ ⌜ ⌝ → ∀ ∗ ∗ ∗ ≡ → (3.35) Z0 (ψ( σ /x0) |xk (f (s , s ) {zxk ψ(xk/x0))}). diag-10 =σ Thus

(3.36) Z0 ⊢ (ψ(⌜σ⌝/x0) → σ). diag-11

(Theorem 3.13)

In the rest of the present subsection, we shall examine several applications of the Diagonal Lemma. We already saw that Tarski’s Undeﬁnability of Truth is a direct consequence of the Diagonal Lemma (Corollary 3.3). We also have the following semantic version of Corol- lary 3.3. For L and T as in Corollary 3.3, a predicate “V |= ·” expressing “ · holds in the universe V” should satisfy at least the following two properties:

27 ′ ′ ′ (3.37) For any extension T of T in L, if T ⊢ φ, then T ⊢ “V |= ⌜φ⌝”; incompl-3-0

(3.38) T ⊢ ∀φ ε FmlL (“V |= ¬φ” → ¬“V |= φ”). incompl-3-1

Theorem 3.14 (Semantic Version of the Undeﬁnability of Truth) Let L and T be as in T-incompl-0-1 Corollary 3.3. Then there is no L-formula χ(x) such that

(3.37′) For some extension T ′ of T in L, if T ′ ⊢ φ, then T ′ ⊢ χ(⌜φ⌝); ′ (3.38 ) T ⊢ ∀φ ε FmlL (χ(¬φ) → ¬χ(φ)).

Proof. Suppose, toward a contradiction, that there is a L-fromula χ(x) with (3.37′) and (3.38′). Without loss of generality, we may assume that T ′ in (3.37′) is identical with T . Let ψ be the formula ¬χ, and let φ be a ﬁxed point of ψ. That is, φ is an L-sentence with

(3.39) T ⊢ φ ↔ ¬χ(⌜φ⌝). incompl-3-2

Claim 3.14.1 T ⊢ φ.

⊢ T, ¬φ ⊢ χ(⌜¬φ⌝) by (3.37′). Thus, T, ¬φ ⊢ ¬χ(⌜φ⌝) by (3.38′) and by Deduction Theorem (Theorem 2.3). On the other hand, we also have T, ¬φ ⊢ χ(⌜φ⌝) by (3.39) and by Dedcuction Theo- rem. Thus, by Lemma 2.4, and again by Dedcuction Theorem, we have T ⊢ (¬φ → (χ(⌜φ⌝) ∧ ¬χ(⌜φ⌝))). Since (¬φ → (χ(⌜φ⌝) ∧ ¬χ(⌜φ⌝))) → φ, is a tautology, it follows that T ⊢ φ. ⊣ (Claim 3.14.1)

Now by Claim 3.14.1 and (3.37′), we have T ⊢ χ(⌜φ⌝). On the other hand, by Claim 3.14.1 and (3.39), we have also T ⊢ ¬χ(⌜φ⌝). This is a contradiction to the assumption that T is consistent. (Theorem 3.14)

Scan 2020-04-26--13.37-HH-.pdf p.16 を参照． L ≺ L if 0# exists! 0# defines the truth in L

The next theorem can be proved using Representability Theorem (Theorem 3.1) and Diagonal Lemma (Theorem 3.2).

T-incompl-1 Theorem 3.15 Suppose that T is a concretely given consistent theory in a language L extending L{} with T ⊇ Z0. Then T is not decidable, that is, Th(T ) = {φ : φ is an L-sentence and T ⊢ φ} is not recursive.

Proof. Assume, toward a contradiction, that T is a concretely given consistent theory in a language L such that T ⊇ Z0 and Th(T ) is decidable. By the Representability

Theorem 3.1, there is an L{}-formula ψ = ψ(x0) such that, for any L-sentence σ,

(3.40) if σ ∈ Th(T ) then T ⊢ ψ(⌜σ⌝) and incompl-4

28 (3.41) if σ ̸∈ Th(T ) then T ⊢ ¬ψ(⌜σ⌝). incompl-5

By the Diagonal Lemma (Theorem 3.2), there is an L{}-sentence σ0 such that

(3.42) T ⊢ σ0 ↔ ¬ψ(⌜σ0⌝). incompl-6

Claim 3.15.1 T ̸⊢ σ0.

⊢ Suppose T ⊢ σ0. By (3.42), it follows that T ⊢ ¬ψ(⌜σ0⌝). Since T is assumed to be consistent, we have T ̸⊢ ψ(⌜σ0⌝). By (3.40), it follows that T ̸⊢ σ0. This is a contradiction. ⊣ (Claim 3.15.1)

By Claim 3.15.1 and by (3.41), it follows that T ⊢ ¬ψ(⌜σ0⌝). Thus, by (3.42), T ⊢ σ0.

This is a contradiction to Claim 3.15.1. (Theorem 3.15)

Corollary 3.16 (A version of the First Incompleteness Theorem) Suppose that T is a T-incompl-2 concretely given consistent theory in a language L extending L{} with T ⊇ Z0. Then T is not complete.

Proof. If T were complete then Th(T ) would be recursive (By enumerating the proofs in the language L of T as P0, P1,..., we can decide whether φ is in Th(T ) or not by checking each these proofs one by one if it is a proof of φ or ¬φ). This would be a contradiction to Theorem 3.15. (Corollary 3.16)

Note that the Corollary above assumes only the consistency of the theory T , while Gödel’soriginal proof of the First Incompleteness Theorem needed the ω-consistency of the theory. To reduce the assumtipn to the consistency of T , Rosser’s trick is necessary if we want to stick to Gödel’soriginal idea of the proof. On the other hand, the Gödel- Rosser proof of the First Incompleteness Theorem is better than the one given above in that the proof gives a concrete example of an independent sentence over T which, intuitively, should be “true”.

3.3 G¨odel’sSpeed-up Theorem and Incompleteness Theorems G-speed-up L L ⊆ { }× logic-20-2019-07- In Z0, let proof (x0, x1, x2) be the {}-formula: “x0 is a theory in a language 4, 5, 6 29.pdf ∗ ω, x2 is an L-formula and x1 is a proof of x2 from x0 in the deduction system K ”.

Let consis(x0) be the L{}-formula: ¬∃x1proof (x0, x1, ⌜x ̸≡ x⌝).

For a concretely given theory T , ⌜⌜T⌝⌝ is not a term in the language of Z0. We merely introduce an abbreviation “x ε ⌜⌜T⌝⌝” of a corresponding formula which should satisfy:

(3.43) For all concretely given formula φ, if φ belongs to T , then we have Z0 ⊢ ⌜φ⌝ ε diag-12

⌜⌜T⌝⌝; if φ does not belong to T , then Z0 ⊢ ⌜φ⌝ ̸ε ⌜⌜T⌝⌝.

29 The expressions proof (⌜⌜T⌝⌝) and consis(⌜⌜T⌝⌝) are to be understood as abbreviations of L{}-formulas expressing “for all x0, if x0 = ⌜⌜T⌝⌝ then proof (x0, x1, x2)” and “for all x0, if x0 = ⌜⌜T⌝⌝ then consis(x0)”, respectively.

T-G-speed-up-0

Theorem 3.17 (A variant of G¨odel’sSpeed-up Theorem) Let T ⊇ Z0 be a concretely given consistent theory in a ﬁnite language L ⊇ L{}. For any recursive function f : N →

N there is an L{}-formula φ = φ(x1) satisfying:

P (3.44) For any n ∈ N, T ⊢ φ(n/x1). For any proof P in L, if T ⊢ φ(n/x1), then G-speed-up-0 L(P) ≥ f(n).

(3.45) T, consis(⌜⌜T⌝⌝) ⊢ (∀x1 ε ω)φ. G-speed-up-1

P∗ Remarks. ( 1 ) If T, consis(⌜⌜T⌝⌝) ⊢ (∀x1 ε ω)φ, then there are proofs Pn, n ∈ N such Pn ∗ that T, consis(⌜⌜T⌝⌝) ⊢ φ(n/x1) and L(Pn) ≤ p(L(P ), n) where p is a polynomials of low degree (1 or 2 depending on the proof system. Exercise: ﬁnd optimal p for our deduction system K∗). Thus, for φ in Theorem 3.17 for fast growing f, we do have a tremendous spead-up, if we go from T to T + consis(⌜⌜T⌝⌝).

( 2 ) Similarly to footnote (36) , any theory concretely given T with T ⊇ Z0 has an ′ mutually translatable theory T ⊇ Z0 in a ﬁnite language.

Proof of Theorem 3.17. Let ψ = ψ(x0, x1) be the L{}-formula: ∈ ∧ ∀ → ¬ ⌜⌜ ⌝⌝ ⌞ ⌟ (3.46) x0 FmlL{} p(L(p) < f(x1) proof ( T , p, Subst(x0, 1, x1 )). G-speed-up-2

By Diagonal Lemma (Theorem 3.13), there is an L{}-formula φ = φ(x1) such that

(3.47) Z0 ⊢ φ ↔ ψ(⌜φ⌝/x0). G-speed-up-3

We show that this φ is as desired.

Cl-G-speed-up-0

Claim 3.17.1 For each n ∈ N, we have T ⊢ φ(n/x1).

⊢ Suppose otherwise and assume that T ̸⊢ φ(n/x1) for some n ∈ N. Then, for any P proof P in L, we have T ̸⊢ φ(n/x1). In particular, this holds for all proofs P in L with L(P) < f(n). Since there are only ﬁnitely many such P (upto the renaming of variable symbols), it follows that

(3.48) Z0 ⊢ ∀p (L(p) < f(n) → ¬proof (⌜⌜T⌝⌝, p, Subst(⌜φ⌝, 1, ⌞n⌟)). G-speed-up-4

Note that Z0 ⊢ Subst(⌜φ⌝, 1, ⌞n⌟) ≡ ⌜φ(n/x1)⌝ by Lemma 3.12, ( 5 ). Thus, by (3.46) and (3.47), it follows that Z0 ⊢ φ(n, x1). This is a contradiction. ⊣ (Claim 3.17.1)

Cl-G-speed-up-1 Claim 3.17.2 For each n ∈ N, there is no proof P in L with L(P) < f(n) such that P T ⊢ φ(n/xn).

⊢ Suppose that there is some n ∈ N and a proof P such that L(P) < f(n) but

30 P (3.49) T ⊢ φ(n/x1). G-speed-up-5

We have

(3.50) Z0 ⊢ “L(⌜P⌝) < f(n)” ∧ proof (⌜⌜T⌝⌝, ⌜P⌝, ⌜φ(n)⌝). G-speed-up-6

By Lemma 3.12, ( 5 ),

(3.51) Z0 ⊢ Subst(⌜φ⌝, 1, ⌞n⌟) ≡ ⌜φ(n/x1)⌝. G-speed-up-7

Thus

(3.52) Z0 ⊢ ¬ψ(⌜φ⌝/x0, n/x1). G-speed-up-8

On the other hand, (3.49) and (3.47) imply Z0 ⊢ ψ(⌜φ⌝/x0, n/x1). Since T ⊇ Z0 this is a contradiction to the assumption of the consistency of T . ⊣ (Claim 3.17.2)

Cl-G-speed-up-2

Claim 3.17.3 T + consis(⌜⌜T⌝⌝) ⊢ (∀x1 ε ω)φ.

⊢ In T + consis(⌜⌜T⌝⌝), we can translate the meta-mathematical argument of Claim ∗ 3.17.1 into a formal proof in K . ⊣ (Claim 3.17.3)

(Theorem 3.17)

Theorem 3.18 (The Second Incompleteness Theorem) Suppose that T ⊇ Z0 is a con- T-G-speed-up-1 cretely given consistent thoery. Then we have T ̸⊢ consis(⌜⌜T⌝⌝).

Proof. We may assume that the language L of T is ﬁnite.(36) Let f : N → N be a fast growing function (e.g. faster than any polynomial function) and let φ be as in Theorem 3.17 for T and this f.

If T ⊢ consis(⌜⌜T⌝⌝), then T ⊢ (∀x1 ∈ ω)φ by (3.45). By Remark ( 1 ), this is a contradiction to (3.44). (Theorem 3.18)

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